That was John Lennon. The melody of this part of the song was later borrowed by Deep Purple for their song "Hush."
Paul McCartney wrote the part of the song that had the lyrics:- What are the parts written by Paul?
Woke up, got out of bed.
Dragged a comb across my head.
Found my way downstairs and drank a cup,
And looking up, I noticed I was late.
Found my coat and grabbed my hat.
Made the bus in seconds flat.
Found my way upstairs and had a smoke.
Somebody spoke and I went into a dream.
What happened with "A Day in the Life" was that John Lennon had written the beginning and ending of a song, but had no middle for it. Paul McCartney had written the middle of a song, but had no beginning or ending for it. In the end, it was decided to fuse the parts that they had into one song, which became "A Day in the Life" as we know it.
It was McCartney's idea to join the two sections of the song with the 24-bar glissando played by a 90-piece orchestra.